Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

TL;DR

This paper studies the structure of Hurwitz moduli varieties that parametrize Galois covers of algebraic curves, establishing conditions under which these spaces are fine or coarse moduli varieties and explicitly constructing universal families.

Contribution

It proves that Hurwitz spaces are fine moduli varieties for trivial center groups and coarse for arbitrary groups, and constructs universal families explicitly.

Findings

Hurwitz space $H^G_n(Y)$ is a fine moduli variety when $G$ has trivial center.

Hurwitz space $H^G_n(Y)$ is a coarse moduli variety for arbitrary $G$.

Explicit universal families are constructed for pointed Galois covers.

Abstract

Given a smooth, projective curve $Y$, a finite group $G$ and a positive integer $n$ we study smooth, proper families $X\to Y\times S\to S$ of Galois covers of $Y$ with Galois group isomorphic to $G$ branched in $n$ points, parameterized by algebraic varieties $S$. When $G$ is with trivial center we prove that the Hurwitz space $H^G_n(Y)$ is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary $G$ we prove that $H^G_n(Y)$ is a coarse moduli variety. For families of pointed Galois covers of $(Y,y_0)$ we prove that the Hurwitz space $H^G_n(Y,y_0)$ is a fine moduli variety, and construct explicitly the universal family, for arbitrary group $G$. We use classical tools of algebraic topology and of complex algebraic geometry.